
Reducing Topological Minor Containment to the Unique Linkage Theorem
In the Topological Minor Containment problem (TMC) problem two undirecte...
read it

A tight ErdősPósa function for planar minors
Let H be a planar graph. By a classical result of Robertson and Seymour,...
read it

Spanning Tree Congestion and Computation of Generalized GyőriLovász Partition
We study a natural problem in graph sparsification, the Spanning Tree Co...
read it

Minimal Separators in Graphs
The Known Menger's theorem states that in a finite graph, the size of a ...
read it

Algorithmic Extensions of Dirac's Theorem
In 1952, Dirac proved the following theorem about long cycles in graphs ...
read it

LowCongestion Shortcuts for Graphs Excluding Dense Minors
We prove that any nnode graph G with diameter D admits shortcuts with c...
read it

A more accurate view of the Flat Wall Theorem
We introduce a supporting combinatorial framework for the Flat Wall Theo...
read it
Quickly excluding a nonplanar graph
A cornerstone theorem in the Graph Minors series of Robertson and Seymour is the result that every graph G with no minor isomorphic to a fixed graph H has a certain structure. The structure can then be exploited to deduce farreaching consequences. The exact statement requires some explanation, but roughly it says that there exist integers k,n depending on H only such that 0<k<n and for every n× n grid minor J of G the graph G has a a knear embedding in a surface Σ that does not embed H in such a way that a substantial part of J is embedded in Σ. Here a knear embedding means that after deleting at most k vertices the graph can be drawn in Σ without crossings, except for local areas of nonplanarity, where crossings are permitted, but at most k of these areas are attached to the rest of the graph by four or more vertices and inside those the graph is constrained in a different way, again depending on the parameter k. The original and only proof so far is quite long and uses many results developed in the Graph Minors series. We give a proof that uses only our earlier paper [A new proof of the flat wall theorem, J. Combin. Theory Ser. B 129 (2018), 158–203] and results from graduate textbooks. Our proof is constructive and yields a polynomial time algorithm to construct such a structure. We also give explicit constants for the structure theorem, whereas the original proof only guarantees the existence of such constants.
READ FULL TEXT
Comments
There are no comments yet.